Overview
This lecture covers how to calculate standard deviation, variance, and range by hand and with a calculator, emphasizing their meanings and differences as measures of data spread.
Calculating Standard Deviation by Hand
- The standard deviation formula involves subtracting each value from the mean, squaring the result, summing all squared differences, dividing by n-1, then taking the square root.
- List all data values and the mean (xĢ) in a table to organize calculations.
- Subtract the mean from each value, square each result, and add them up to get the sum of squared differences.
- Divide this sum by (n-1), where n is the sample size, for the variance.
- Take the square root of the variance to get the standard deviation.
- Always round the final answer to one decimal place beyond the original data's precision.
Using a Calculator for Standard Deviation
- Input data into the calculator under "STAT" and "EDIT" menus (typically into list L1).
- To clear old data, use the arrow to highlight L1, then select "Clear" and "Enter."
- Use "STAT" ā "CALC" ā "1-Var Stats" and specify the list to have the calculator compute mean and standard deviation.
- The standard deviation (Sx) shown matches hand-calculated results.
Comparing Measures of Variation
- Range is found by subtracting the minimum value from the maximum value.
- Standard deviation reflects the typical distance of values from the mean.
- A larger standard deviation or range indicates more spread in the data.
Variance (sĀ²)
- Variance is the square of the standard deviation (sĀ² = (Sx)Ā²).
- To find variance, use the unrounded standard deviation value; otherwise, rounding first may introduce error.
- Variance offers another numerical measure of data spread but is less commonly reported than standard deviation.
Key Terms & Definitions
- Mean (xĢ) ā the average of all data values.
- Standard Deviation (Sx) ā the typical distance data values are from the mean, calculated as the square root of variance.
- Variance (sĀ²) ā the average squared distance from each data value to the mean.
- Range ā the difference between the maximum and minimum values in a dataset.
- Sample Size (n) ā the number of data points in the sample.
Action Items / Next Steps
- Practice calculating standard deviation, variance, and range both by hand and with a calculator.
- Read the top of page 29 on variance and complete related homework problems.
- Prepare to discuss additional measures of variation in the next class.